A Learning Environment for Mathematics

Martin Homik, Erica Melis-- ActiveMath Group –

ABIS 2004, Berlin

German Research Center for Artificial IntelligenceDFKI GmbH, Saarbrücken

Outline

• Features• Examples• Key components:

– Knowledge representation– User model– Course generator

• Conclusion

• Adaptive course generation• Adaptive presentation• Semantic knowledge representation• User model• Pedagogical knowledge• Interactive exercises • Suggestion mechanism• …

ActiveMath‘s Features

Different users – one topic

Anton• Mathematics• Bachelor student

• Group theory• Exam exercises• Train interactively• At home

Eva• Computer science• PhD student

• Group theory• Profound knowledge• Overview • At university

„Eva‘s Books“

„A Book“

Good masteryGood mastery

Medium masteryMedium mastery

Weak masteryWeak mastery

„A Book“

Scenario: Overview

Scenario: Exam

Scenario: No sufficient knowledge

Scenario: No sufficient knowledge

Adaptivity: Field

Mathematics

Biology

Physics

Knowledge Representation

Extended OMDoc • Extension of OpenMath• Markup format for mathematical

documents• Content items (concept, additional)• Metadata

– Difficulty– Abstractness– Field– Learning context– …

Knowledge RepresentationContent items• unique ID

Content items• unique ID

Concepts Additional

Definitions Assertions

ProofAxioms

Examples

Exercises Motivation

Elaboration

Relations:• Mathematical dependency• Pedagogical prerequisite

Relations:• Pedagogical dependency

User Model

History Profile

A1 A2 A3 A4 A5

• Content IDs• Start/End timestamp • Rate of success (exercise)• Type:

• read• example• exercise• self assessment

• Preferences:• Language• Field • Gender• Scenario• Goals

• Mastery (per concept):• Knowledge• Comprehension• Application

• Justification

User Model: Justification

A1 A2 A3 A4 A5

Concept:• Concept ID• Mastery:

• Knowledge (0.3)• Comprehension (0.1)• Application (0.1)

• Justification• History Step 4• History Step 2

Concept:• Concept ID• Mastery:

• Knowledge (0.5)• Comprehension (0.3)• Application (0.1)

• Justification• History Step 5• History Step 4

History

Profile (mastery)

User Model: Update

Trigger (Action):• Page changed• Exercise finished

Evaluator (page):• fetch list of page items• compute values:

• knowledge• comprehension

User Model:• update

Evaluator (exercise):• compute application value:

• difficulty• sucess rate

Software Eyetracker (DFKEye)

• Trace attention on regions• More fine-grained mastery evaluation

Course Generation

User ModelUser Model

mBasemBase PedagogicalRules

PedagogicalRules

CourseGenerator

CourseGenerator

Course Generation (1)

Goal concept

1. Retrieve content from knowledge base• Start with goal concept• Collect recursively concepts the goal depends on• … plus additional information (e.g. examples)

Course Generation (2)

Goal concept

2. Filter concepts• According to pedagogical rules• According to the user model

Defrule PatternExamPrepAllow (definition, exercise)Order (definition, exercise)

Defrule ReqAppExAllow (definition, exercise)Test (user-kb(definition)<0.3)Add ex-for(definition 0.3) ex-for(definition 0.5)

Course Generation (3)

Goal concept

3. Linearize graph • according to mathematical dependency• according to pedagogical dependency

Architecture

mBasemBase

PedagogicalRules

PedagogicalRules

WebServer

WebServer

SessionManager

SessionManager

MathSystems

MathSystems

CourseGenerator

CourseGenerator

PresentationGenerator

(XSLT)

PresentationGenerator

(XSLT)

User Model

HistoryHistoryProfileProfile

XML-RPC

Java

http

http

LeActiveMath

• Tutorial dialogues• Data base of interactive exercises• Learning-effective tools • Extend knowledge representation• Improve open user model• Classroom tests

Conclusion www.activemath.org

• Stable!!!• Currently available content:

– Analysis– Algebra– Optimization– Operation research– Statistics– …

• Internationalization• Employed at school and at university